Thermal Physics, Homework #1 problem #1

Consider a large number N of localized particles in an external magnetic field H. Each particle has spin 1/2.

Find the number of states, g(N,M), accessible to the system as a function of M=(Nup-Ndown), the magnetization.

Calculate the entropy per particle.

Determine the value of M for which the number of states is a maximum for a given N.

Equations that may help?
N=Nup+Ndown

M=Nup-Ndown

g(N,s)= N!/(Nup!Ndown!)

σ(N,U)=log(g(N,U))

This is my first thermal physics course and I am kinda confused (and overwhelmed) by this first homework assignment if anyone could explain what I am suposed to do, or set me in a direction, I would appreciate it.

"Determine the value of M for which the number of states is a maximum for a given N."
The value of M should be 0 correct?, because the middle of the Gaussian distribution will be centered at the origin. M=Nup-Ndown

"Determine the value of M for which the number of states is a maximum for a given N."
The value of M should be 0 correct?, because the middle of the Gaussian distribution will be centered at the origin. M=Nup-Ndown

Correct, but I would say this as "the value of M will be zero because the maximum of the Gaussian distribution will be when [itex]N_{up}=N_{down}=N/2\rightarrow M=N_{up}-N_{down}=N_{up}-N_{up}=0[/itex]" rather than how you have it.